ab-angle->ABCF A

Percentage Accurate: 79.4% → 79.4%

Time: 16.3s

Alternatives: 14

Speedup: N/A×

• 36.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (angle / 180.0) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle / 180.0) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (angle / 180.0) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle / 180.0) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 79.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(angle\_m \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 2e-5)
   (+
    (* b b)
    (pow
     (*
      angle_m
      (*
       a
       (*
        PI
        (fma
         (* (* angle_m angle_m) -2.8577960676726107e-8)
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (fma
    (* a (fma (cos (* PI (* angle_m 0.011111111111111112))) -0.5 0.5))
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-5) {
		tmp = (b * b) + pow((angle_m * (a * (((double) M_PI) * fma(((angle_m * angle_m) * -2.8577960676726107e-8), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = fma((a * fma(cos((((double) M_PI) * (angle_m * 0.011111111111111112))), -0.5, 0.5)), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-5)
		tmp = Float64(Float64(b * b) + (Float64(angle_m * Float64(a * Float64(pi * fma(Float64(Float64(angle_m * angle_m) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = fma(Float64(a * fma(cos(Float64(pi * Float64(angle_m * 0.011111111111111112))), -0.5, 0.5)), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-5], N[(N[(b * b), $MachinePrecision] + N[Power[N[(angle$95$m * N[(a * N[(Pi * N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000016e-5

   1. Initial program 88.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\color{blue}{b}}^{2} \]

      2. pow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]

      3. lift-*.f64 88.4

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   7. Applied rewrites 88.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   8. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\color{blue}{\left(\frac{-1}{34992000} \cdot a\right) \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)} + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + b \cdot b \]

      2. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot a\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)} + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + b \cdot b \]

      3. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\color{blue}{\left(\left(\frac{-1}{34992000} \cdot a\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}} + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + b \cdot b \]

      4. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\color{blue}{\left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot {angle}^{2} + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + b \cdot b \]

      5. +-commutative N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)}\right)}^{2} + b \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}}^{2} + b \cdot b \]

      7. associate-*l* N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\frac{-1}{34992000} \cdot \left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right)\right)}^{2} + b \cdot b \]

      8. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \color{blue}{\left(a \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)\right)}\right)\right)}^{2} + b \cdot b \]

      9. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \left(a \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right)\right)}^{2} + b \cdot b \]

   10. Applied rewrites 83.1%

       \[\leadsto {\color{blue}{\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}}^{2} + b \cdot b \]

   if 2.00000000000000016e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.1%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   7. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), -0.5, 0.5\right), a, b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\right)}^{2} + b \cdot b \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (* (/ angle_m 180.0) PI))) 2.0) (* b b)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((angle_m / 180.0) * ((double) M_PI)))), 2.0) + (b * b);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin(((angle_m / 180.0) * Math.PI))), 2.0) + (b * b);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin(((angle_m / 180.0) * math.pi))), 2.0) + (b * b)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(Float64(angle_m / 180.0) * pi))) ^ 2.0) + Float64(b * b))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin(((angle_m / 180.0) * pi))) ^ 2.0) + (b * b);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
4. Step-by-step derivation

5. Applied rewrites 81.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
6. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\color{blue}{b}}^{2} \]

   2. pow2 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   3. lift-*.f64 81.0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

7. Applied rewrites 81.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]
8. Add Preprocessing

Alternative 3: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b + {\left(a \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (* b b) (pow (* a (sin (* (* angle_m PI) 0.005555555555555556))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return (b * b) + pow((a * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return (b * b) + Math.pow((a * Math.sin(((angle_m * Math.PI) * 0.005555555555555556))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return (b * b) + math.pow((a * math.sin(((angle_m * math.pi) * 0.005555555555555556))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(Float64(b * b) + (Float64(a * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (b * b) + ((a * sin(((angle_m * pi) * 0.005555555555555556))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
4. Step-by-step derivation

5. Applied rewrites 81.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
6. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\color{blue}{b}}^{2} \]

   2. pow2 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   3. lift-*.f64 81.0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

7. Applied rewrites 81.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]
8. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + b \cdot b \]

   2. associate-*l/ N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \mathsf{PI}\left(\right)}{180}\right)}\right)}^{2} + b \cdot b \]

   3. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}{180}\right)\right)}^{2} + b \cdot b \]

   4. div-inv N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + b \cdot b \]

   5. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + b \cdot b \]

   6. metadata-eval 80.7

      \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.005555555555555556}\right)\right)}^{2} + b \cdot b \]

9. Applied rewrites 80.7%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)}^{2} + b \cdot b \]

10. Final simplification 80.7%

    \[\leadsto b \cdot b + {\left(a \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}^{2} \]
11. Add Preprocessing

Alternative 4: 79.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 2e-5)
   (+ (* b b) (pow (* a (* angle_m (* PI 0.005555555555555556))) 2.0))
   (fma
    (* a (fma (cos (* PI (* angle_m 0.011111111111111112))) -0.5 0.5))
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-5) {
		tmp = (b * b) + pow((a * (angle_m * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = fma((a * fma(cos((((double) M_PI) * (angle_m * 0.011111111111111112))), -0.5, 0.5)), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-5)
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(angle_m * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = fma(Float64(a * fma(cos(Float64(pi * Float64(angle_m * 0.011111111111111112))), -0.5, 0.5)), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-5], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000016e-5

   1. Initial program 88.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\color{blue}{b}}^{2} \]

      2. pow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]

      3. lift-*.f64 88.4

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   7. Applied rewrites 88.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   8. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{180}\right)}}^{2} + b \cdot b \]

      2. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}}^{2} + b \cdot b \]

      3. associate-*r* N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)}\right)}^{2} + b \cdot b \]

      4. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + b \cdot b \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      8. lower-PI.f64 84.1

         \[\leadsto {\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \color{blue}{\pi}\right)\right)\right)}^{2} + b \cdot b \]

   10. Applied rewrites 84.1%

       \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}}^{2} + b \cdot b \]

   if 2.00000000000000016e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.1%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   7. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), -0.5, 0.5\right), a, b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 2e-5)
   (+ (* b b) (pow (* a (* angle_m (* PI 0.005555555555555556))) 2.0))
   (fma
    (* a (fma (cos (* angle_m (* PI 0.011111111111111112))) -0.5 0.5))
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-5) {
		tmp = (b * b) + pow((a * (angle_m * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = fma((a * fma(cos((angle_m * (((double) M_PI) * 0.011111111111111112))), -0.5, 0.5)), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-5)
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(angle_m * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = fma(Float64(a * fma(cos(Float64(angle_m * Float64(pi * 0.011111111111111112))), -0.5, 0.5)), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-5], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Cos[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000016e-5

   1. Initial program 88.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\color{blue}{b}}^{2} \]

      2. pow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]

      3. lift-*.f64 88.4

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   7. Applied rewrites 88.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   8. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{180}\right)}}^{2} + b \cdot b \]

      2. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}}^{2} + b \cdot b \]

      3. associate-*r* N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)}\right)}^{2} + b \cdot b \]

      4. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + b \cdot b \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      8. lower-PI.f64 84.1

         \[\leadsto {\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \color{blue}{\pi}\right)\right)\right)}^{2} + b \cdot b \]

   10. Applied rewrites 84.1%

       \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}}^{2} + b \cdot b \]

   if 2.00000000000000016e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.1%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\color{blue}{b}}^{2} \]

      2. pow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]

      3. lift-*.f64 62.1

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   7. Applied rewrites 62.1%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   9. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{-133}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 8.2e-133)
   (* (* b b) (fma 0.5 (cos (* (* angle_m PI) 0.011111111111111112)) 0.5))
   (+ (* b b) (pow (* a (* angle_m (* PI 0.005555555555555556))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 8.2e-133) {
		tmp = (b * b) * fma(0.5, cos(((angle_m * ((double) M_PI)) * 0.011111111111111112)), 0.5);
	} else {
		tmp = (b * b) + pow((a * (angle_m * (((double) M_PI) * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 8.2e-133)
		tmp = Float64(Float64(b * b) * fma(0.5, cos(Float64(Float64(angle_m * pi) * 0.011111111111111112)), 0.5));
	else
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(angle_m * Float64(pi * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 8.2e-133], N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.20000000000000045e-133

   1. Initial program 79.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{e^{\log a}} \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. lift-PI.f64 N/A

         \[\leadsto {\left(e^{\log a} \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. lift-/.f64 N/A

         \[\leadsto {\left(e^{\log a} \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(e^{\log a} \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lift-sin.f64 N/A

         \[\leadsto {\left(e^{\log a} \cdot \color{blue}{\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log a}\right)}^{2} \cdot {\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. rem-exp-log N/A

         \[\leadsto {\color{blue}{a}}^{2} \cdot {\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{{\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {a}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {a}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right) \cdot \left(a \cdot a\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), -0.5, 0.5\right), a \cdot a, b \cdot \left(b \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), 0.5\right)\right)\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{1}{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{90}\right), \frac{1}{2}\right) \]

      10. lower-PI.f64 57.7

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \color{blue}{\pi}\right) \cdot 0.011111111111111112\right), 0.5\right) \]

   9. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)} \]

   if 8.20000000000000045e-133 < a

   1. Initial program 83.2%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 83.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\color{blue}{b}}^{2} \]

      2. pow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]

      3. lift-*.f64 83.3

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   7. Applied rewrites 83.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   8. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{180}\right)}}^{2} + b \cdot b \]

      2. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}}^{2} + b \cdot b \]

      3. associate-*r* N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)}\right)}^{2} + b \cdot b \]

      4. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + b \cdot b \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      8. lower-PI.f64 80.6

         \[\leadsto {\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \color{blue}{\pi}\right)\right)\right)}^{2} + b \cdot b \]

   10. Applied rewrites 80.6%

       \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}}^{2} + b \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{-133}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{-133}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 8.2e-133)
   (* (* b b) (fma 0.5 (cos (* angle_m (* PI 0.011111111111111112))) 0.5))
   (+ (* b b) (pow (* a (* angle_m (* PI 0.005555555555555556))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 8.2e-133) {
		tmp = (b * b) * fma(0.5, cos((angle_m * (((double) M_PI) * 0.011111111111111112))), 0.5);
	} else {
		tmp = (b * b) + pow((a * (angle_m * (((double) M_PI) * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 8.2e-133)
		tmp = Float64(Float64(b * b) * fma(0.5, cos(Float64(angle_m * Float64(pi * 0.011111111111111112))), 0.5));
	else
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(angle_m * Float64(pi * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 8.2e-133], N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[Cos[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.20000000000000045e-133

   1. Initial program 79.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{e^{\log a}} \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. lift-PI.f64 N/A

         \[\leadsto {\left(e^{\log a} \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. lift-/.f64 N/A

         \[\leadsto {\left(e^{\log a} \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(e^{\log a} \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lift-sin.f64 N/A

         \[\leadsto {\left(e^{\log a} \cdot \color{blue}{\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log a}\right)}^{2} \cdot {\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. rem-exp-log N/A

         \[\leadsto {\color{blue}{a}}^{2} \cdot {\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{{\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {a}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {a}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right) \cdot \left(a \cdot a\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), -0.5, 0.5\right), a \cdot a, b \cdot \left(b \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), 0.5\right)\right)\right)} \]

   8. Applied rewrites 65.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5, 0.5\right) - \mathsf{fma}\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right)}, -0.5, 0.5\right), a \cdot a, b \cdot \left(b \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), 0.5\right)\right)\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2}\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right)} \]

       6. lower-cos.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{1}{2}\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right), \frac{1}{2}\right) \]

       11. lower-PI.f64 58.3

           \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\color{blue}{\pi} \cdot 0.011111111111111112\right)\right), 0.5\right) \]

   11. Applied rewrites 58.3%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)} \]

   if 8.20000000000000045e-133 < a

   1. Initial program 83.2%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 83.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\color{blue}{b}}^{2} \]

      2. pow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]

      3. lift-*.f64 83.3

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   7. Applied rewrites 83.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   8. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{180}\right)}}^{2} + b \cdot b \]

      2. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}}^{2} + b \cdot b \]

      3. associate-*r* N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)}\right)}^{2} + b \cdot b \]

      4. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + b \cdot b \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      8. lower-PI.f64 80.6

         \[\leadsto {\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \color{blue}{\pi}\right)\right)\right)}^{2} + b \cdot b \]

   10. Applied rewrites 80.6%

       \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}}^{2} + b \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{-133}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.9e-17)
   (* b b)
   (+ (* b b) (pow (* a (* angle_m (* PI 0.005555555555555556))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.9e-17) {
		tmp = b * b;
	} else {
		tmp = (b * b) + pow((a * (angle_m * (((double) M_PI) * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.9e-17) {
		tmp = b * b;
	} else {
		tmp = (b * b) + Math.pow((a * (angle_m * (Math.PI * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.9e-17:
		tmp = b * b
	else:
		tmp = (b * b) + math.pow((a * (angle_m * (math.pi * 0.005555555555555556))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.9e-17)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(angle_m * Float64(pi * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.9e-17)
		tmp = b * b;
	else
		tmp = (b * b) + ((a * (angle_m * (pi * 0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.9e-17], N[(b * b), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.9000000000000001e-17

   1. Initial program 77.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 58.5

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 1.9000000000000001e-17 < a

   1. Initial program 88.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\color{blue}{b}}^{2} \]

      2. pow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{b \cdot b} \]

      3. lift-*.f64 88.4

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   7. Applied rewrites 88.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   8. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{180}\right)}}^{2} + b \cdot b \]

      2. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}}^{2} + b \cdot b \]

      3. associate-*r* N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)}\right)}^{2} + b \cdot b \]

      4. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + b \cdot b \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2} + b \cdot b \]

      8. lower-PI.f64 86.7

         \[\leadsto {\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \color{blue}{\pi}\right)\right)\right)}^{2} + b \cdot b \]

   10. Applied rewrites 86.7%

       \[\leadsto {\color{blue}{\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}}^{2} + b \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \left(angle\_m \cdot angle\_m\right) \cdot t\_0, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI PI) 3.08641975308642e-5)))
   (if (<= a 1.9e-17)
     (* b b)
     (if (<= a 1.12e+173)
       (fma (* a a) (* (* angle_m angle_m) t_0) (* b b))
       (* a (* (* a (* angle_m angle_m)) t_0))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5;
	double tmp;
	if (a <= 1.9e-17) {
		tmp = b * b;
	} else if (a <= 1.12e+173) {
		tmp = fma((a * a), ((angle_m * angle_m) * t_0), (b * b));
	} else {
		tmp = a * ((a * (angle_m * angle_m)) * t_0);
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(pi * pi) * 3.08641975308642e-5)
	tmp = 0.0
	if (a <= 1.9e-17)
		tmp = Float64(b * b);
	elseif (a <= 1.12e+173)
		tmp = fma(Float64(a * a), Float64(Float64(angle_m * angle_m) * t_0), Float64(b * b));
	else
		tmp = Float64(a * Float64(Float64(a * Float64(angle_m * angle_m)) * t_0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]}, If[LessEqual[a, 1.9e-17], N[(b * b), $MachinePrecision], If[LessEqual[a, 1.12e+173], N[(N[(a * a), $MachinePrecision] * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.9000000000000001e-17

   1. Initial program 77.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 58.5

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 1.9000000000000001e-17 < a < 1.12e173

   1. Initial program 76.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 76.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]

   if 1.12e173 < a

   1. Initial program 99.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.6%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {a}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {a}^{2}} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \cdot {a}^{2} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \cdot {a}^{2} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       11. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       17. lower-PI.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       18. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right) \]

       19. lower-PI.f64 56.2

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{\pi} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       3. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right)\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot a} \]

       9. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot a} \]

   13. Applied rewrites 71.4%

       \[\leadsto \color{blue}{\left(\left(a \cdot \left(angle \cdot angle\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.7% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot \left(a \cdot a\right), \left(\pi \cdot \pi\right) \cdot \left(angle\_m \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(a \cdot \left(angle\_m \cdot angle\_m\right)\right)\right), \pi \cdot \pi, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.1e+132)
   (fma
    (* angle_m (* a a))
    (* (* PI PI) (* angle_m 3.08641975308642e-5))
    (* b b))
   (fma
    (* 3.08641975308642e-5 (* a (* a (* angle_m angle_m))))
    (* PI PI)
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.1e+132) {
		tmp = fma((angle_m * (a * a)), ((((double) M_PI) * ((double) M_PI)) * (angle_m * 3.08641975308642e-5)), (b * b));
	} else {
		tmp = fma((3.08641975308642e-5 * (a * (a * (angle_m * angle_m)))), (((double) M_PI) * ((double) M_PI)), (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.1e+132)
		tmp = fma(Float64(angle_m * Float64(a * a)), Float64(Float64(pi * pi) * Float64(angle_m * 3.08641975308642e-5)), Float64(b * b));
	else
		tmp = fma(Float64(3.08641975308642e-5 * Float64(a * Float64(a * Float64(angle_m * angle_m)))), Float64(pi * pi), Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.1e+132], N[(N[(angle$95$m * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(angle$95$m * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(3.08641975308642e-5 * N[(a * N[(a * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.09999999999999994e132

   1. Initial program 77.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 77.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) + b \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) + b \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right) + b \cdot b \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + b \cdot b \]

      8. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{b \cdot b} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + b \cdot b \]

      10. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      11. associate-*l* N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + b \cdot b \]

      12. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + b \cdot b \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot angle, angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), b \cdot b\right)} \]

   10. Applied rewrites 71.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot angle, \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right), b \cdot b\right)} \]

   if 1.09999999999999994e132 < a

   1. Initial program 99.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.6%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) + b \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) + b \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right) + b \cdot b \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + b \cdot b \]

      8. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{b \cdot b} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + b \cdot b \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} + b \cdot b \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(angle \cdot angle\right)\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} + b \cdot b \]

      12. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(a \cdot a\right) \cdot \left(angle \cdot angle\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} + b \cdot b \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(angle \cdot angle\right)\right) \cdot \frac{1}{32400}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), b \cdot b\right)} \]

   10. Applied rewrites 86.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(a \cdot a\right), \left(\pi \cdot \pi\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right), \pi \cdot \pi, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.9% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot \left(a \cdot a\right), \left(\pi \cdot \pi\right) \cdot \left(angle\_m \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 5e+181)
   (fma
    (* angle_m (* a a))
    (* (* PI PI) (* angle_m 3.08641975308642e-5))
    (* b b))
   (* a (* (* a (* angle_m angle_m)) (* (* PI PI) 3.08641975308642e-5)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 5e+181) {
		tmp = fma((angle_m * (a * a)), ((((double) M_PI) * ((double) M_PI)) * (angle_m * 3.08641975308642e-5)), (b * b));
	} else {
		tmp = a * ((a * (angle_m * angle_m)) * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 5e+181)
		tmp = fma(Float64(angle_m * Float64(a * a)), Float64(Float64(pi * pi) * Float64(angle_m * 3.08641975308642e-5)), Float64(b * b));
	else
		tmp = Float64(a * Float64(Float64(a * Float64(angle_m * angle_m)) * Float64(Float64(pi * pi) * 3.08641975308642e-5)));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 5e+181], N[(N[(angle$95$m * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(angle$95$m * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000003e181

   1. Initial program 78.2%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 78.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) + b \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) + b \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right) + b \cdot b \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + b \cdot b \]

      8. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{b \cdot b} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + b \cdot b \]

      10. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + b \cdot b \]

      11. associate-*l* N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + b \cdot b \]

      12. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + b \cdot b \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot angle, angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), b \cdot b\right)} \]

   10. Applied rewrites 72.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot angle, \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right), b \cdot b\right)} \]

   if 5.0000000000000003e181 < a

   1. Initial program 99.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.5%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {a}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {a}^{2}} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \cdot {a}^{2} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \cdot {a}^{2} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       11. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       17. lower-PI.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       18. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right) \]

       19. lower-PI.f64 58.6

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{\pi} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right) \]

   11. Applied rewrites 58.6%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       3. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right)\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot a} \]

       9. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot a} \]

   13. Applied rewrites 75.9%

       \[\leadsto \color{blue}{\left(\left(a \cdot \left(angle \cdot angle\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(a \cdot a\right), \left(\pi \cdot \pi\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+159}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.3e+159)
   (* b b)
   (* a (* (* a (* angle_m angle_m)) (* (* PI PI) 3.08641975308642e-5)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.3e+159) {
		tmp = b * b;
	} else {
		tmp = a * ((a * (angle_m * angle_m)) * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.3e+159) {
		tmp = b * b;
	} else {
		tmp = a * ((a * (angle_m * angle_m)) * ((Math.PI * Math.PI) * 3.08641975308642e-5));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.3e+159:
		tmp = b * b
	else:
		tmp = a * ((a * (angle_m * angle_m)) * ((math.pi * math.pi) * 3.08641975308642e-5))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.3e+159)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * Float64(Float64(a * Float64(angle_m * angle_m)) * Float64(Float64(pi * pi) * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.3e+159)
		tmp = b * b;
	else
		tmp = a * ((a * (angle_m * angle_m)) * ((pi * pi) * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.3e+159], N[(b * b), $MachinePrecision], N[(a * N[(N[(a * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.3e159

   1. Initial program 77.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 59.1

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Applied rewrites 59.1%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 1.3e159 < a

   1. Initial program 99.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.6%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {a}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {a}^{2}} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \cdot {a}^{2} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \cdot {a}^{2} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       11. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       17. lower-PI.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       18. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right) \]

       19. lower-PI.f64 58.4

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{\pi} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right) \]

   11. Applied rewrites 58.4%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       3. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right)\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot a} \]

       9. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot a} \]

   13. Applied rewrites 70.5%

       \[\leadsto \color{blue}{\left(\left(a \cdot \left(angle \cdot angle\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+159}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{+148}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(\left(a \cdot a\right) \cdot \left(angle\_m \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 6e+148)
   (* b b)
   (* angle_m (* (* a a) (* angle_m (* (* PI PI) 3.08641975308642e-5))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 6e+148) {
		tmp = b * b;
	} else {
		tmp = angle_m * ((a * a) * (angle_m * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 6e+148) {
		tmp = b * b;
	} else {
		tmp = angle_m * ((a * a) * (angle_m * ((Math.PI * Math.PI) * 3.08641975308642e-5)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 6e+148:
		tmp = b * b
	else:
		tmp = angle_m * ((a * a) * (angle_m * ((math.pi * math.pi) * 3.08641975308642e-5)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 6e+148)
		tmp = Float64(b * b);
	else
		tmp = Float64(angle_m * Float64(Float64(a * a) * Float64(angle_m * Float64(Float64(pi * pi) * 3.08641975308642e-5))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 6e+148)
		tmp = b * b;
	else
		tmp = angle_m * ((a * a) * (angle_m * ((pi * pi) * 3.08641975308642e-5)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 6e+148], N[(b * b), $MachinePrecision], N[(angle$95$m * N[(N[(a * a), $MachinePrecision] * N[(angle$95$m * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.00000000000000029e148

   1. Initial program 77.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 59.1

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Applied rewrites 59.1%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 6.00000000000000029e148 < a

   1. Initial program 99.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.6%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {a}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {a}^{2}} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \cdot {a}^{2} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \cdot {a}^{2} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       11. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       17. lower-PI.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       18. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right) \]

       19. lower-PI.f64 58.4

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{\pi} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right) \]

   11. Applied rewrites 58.4%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       3. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)} \cdot \left(a \cdot a\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \cdot \left(a \cdot a\right) \]

       11. associate-*l* N/A

           \[\leadsto \color{blue}{\left(angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right)} \cdot \left(a \cdot a\right) \]

       12. associate-*l* N/A

           \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \cdot \left(a \cdot a\right)\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \cdot \left(a \cdot a\right)\right)} \]

       14. lower-*.f64 N/A

           \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right) \cdot \left(a \cdot a\right)\right)} \]

   13. Applied rewrites 64.6%

       \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{+148}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.5% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* b b))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return b * b;
}

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = b * b
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return b * b;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return b * b

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(b * b)
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = b * b;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 80.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{b}^{2}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{b \cdot b} \]

   2. lower-*.f64 54.1

      \[\leadsto \color{blue}{b \cdot b} \]

5. Applied rewrites 54.1%

   \[\leadsto \color{blue}{b \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))